Relationship between $\|AB\|_*$ and $\|A\|_*\|B\|_*$

Question

Suppose we have two matrices $A\in \mathbb{R}^{m \times n}$ and $B\in \mathbb{R}^{n \times p}$, then what's the relationship between $\|AB\|_*$ and $\|A\|_*\|B\|_*$? The notation $\|\cdot\|_*$ means the nuclear norm (aka trace norm).

If I need to bound $\|AB\|_*$, what restrictions should be added on $A$ and $B$?

Thanks for your answer!

Answer 1

The nuclear norm is submultiplicative, which is to say that it satisfies $\|AB\|_* \leq \|A\|_* \|B\|_*$.

For a direct proof of this fact, it suffices to note that $$ \|AB\|_* \leq \|A\| \cdot \|B\|_* \leq \|A\|_* \cdot \|B\|_*, $$ where $\|A\|$ denotes the spectral norm.